<!doctype html public "-//W3C//DTD HTML 3.2 Final//EN">
<html>

<head>
<meta HTTP-EQUIV="Content-Type" CONTENT="text/html;charset=utf-8">
<title>Xah: Special Plane Curves: Cassinian Oval</title>
</head>

<body>
<pre>back to <a href="../specialPlaneCurves.html">Table of Contents</a></pre>
<h1><a name="Top">Cassinian Oval</a></h1>

<center>
<img src="cassinianOvalMask.png" alt="" width="685" height="409"><br>
Cassinian Ovals
<p><img src="../Icons_dir/mmaIconSmall.gif" width="16" height="16"> <a href="cassinianOvalMask.nb">Graphics code</a>.</p></center>


<!-- page -->
<table border="1">
<tr>
<TD><a href="#History">History</a></TD>
<TD><a href="#Description">Description</a></TD>
<TD><a href="#Formulas">Formulas</a></TD>
<TD><a href="#Properties">Properties</a></TD>
<TD><a href="#Related%20Web%20Sites">Related Web Sites</a></TD>
</tr>
</table>

<hr>
<h2><a name="History">History</a></h2>

<p>Studied by Giovanni Domenico (1680) in  relation to motions of earth and sun.</p>

<hr>
<h2><a name="Description">Description</a></h2>

<p>Cassinian oval describe a family of curves. Cassinian Oval is defined as follows: Given fixed points F1 and F2. Given a constant c. The locus of points such that distance[P,F1] * distance[P,F2] == c is cassinian oval.</p>


<P>
Cassinian oval is analogous to the definition of <a href="../Ellipse_dir/ellipse.html">ellipse</a>, where sum of two distances is replace by product.
</p>

<p><img src="../Icons_dir/mmaIconSmall.gif" width="16" height="16"> <a href="cassinianOval.nb">definition.</a>.</p>

<p>
The fixed points F1 and F2 are called foci. Let the foci be {a,0} and {-a,0}. Let the constant c be b^2. The distance from a point {x,y} to another {m,n} is Sqrt[(x-m)^2+(y-n)^2]. Thus the equation for cassinian oval is Sqrt[(x-a)^2+y^2]*Sqrt[(x+a)^2+y^2]==b^2
</p>

<p>
When the square root of the constant c is less that half the distance between the foci, then there are two branches of the curve. In other words, if b &lt; a, then there are two ovals. When a==b, the curve is called <a href="../LemniscateOfBernoulli_dir/lemniscateOfBernoulli.html">lemniscate of Bernoulli</a></p>


<center><img src="cassinianOval.png" width="328" height="285"><br>
<img src="../Icons_dir/gspIconSmall.gif" width="14" height="16"> <a href="cassinianOvalFamily.gsp">Cassinian Oval Family</a><br>
<p>Cassinian Ovals with a == 1 and different values of b.</p>
</center>


<hr>
<h2><a name="Formulas">Formulas</a></h2>
<ul>
<li>Parametric: Sqrt[M/2] {Cos[t], Sin[t]}, where M is 2 *a^2* Cos[2 t] + 2 *Sqrt[(-a^4+b^4) + a^4 Cos[2* t]^2], 0 &lt; t &lt;= 2 π, and a &lt; b. (This parametrization does not generate the complete curve when a &gt; b.) <img src="../Icons_dir/gcf.gif" width="12" height="16"> <a href="cassini_oval_parametric.gcf">parametric plot</a></li>
<li>Polar: r^4 + a^4 - 2 * r^2 * a^2 * Cos[2 * θ] == b^4. <img src="../Icons_dir/gcf.gif" width="12" height="16"> <a href="cassinian_oval_polar.gcf">polar plot</a></li>
<li>Bipolar: r1 * r2 == b^2.</li>
<li>Cartesian: Sqrt[(x-a)^2+y^2]*Sqrt[(x+a)^2+y^2]==b^2. <img src="../Icons_dir/gcf.gif" width="12" height="16"> <a href="cassinian_oval_rect.gcf">plot</a>, <a href="cassinian_oval_surface.gcf">contour plot</a>
</li>


</ul>


<!-- on the derivation of the polar or parametric formulas, from Richard Palais, 2004-11:

  I think what happens is that if you substitute  x = r cos(theta) 
and y = r sin(theta) into the implicit equation for the oval you get
a "bi-quadratic" equation for r, i.e., one of the form 

  A r^4 + B r^2 + C = 0

where A, B, and C involve only theta but not r. You can then
solve this equation for r^2 in terms of A,B, and C using the 
quadratic formula, and then just take the square root of that 
expression to get r. 

-->

<hr>
<h2><a name="Properties">Properties</a></h2>

<h3>Torus cut</h3>


<p>Cassinian ovals are the intersection of a torus and a plane in certain positions.</p>

<P>Let c be the radius of the torus tube. Let d be the distance from the center of the tube to axis of the torus. The intersection of a plane c distant from the torus's axis is a Cassinian oval, with a = d and b^2 = Sqrt[4]*c*d, where a is half of the distance between foci, and b^2 is the product constant.</p>

<P>One thing we realize is that for Cassinian oval with large constant b^2, the curve approches a circle, and the corresponding torus is one such that the tube radius is larger than the center to directrix. That is, a self-intersecting torus without the hole. This surface also approaches a sphere.</p>

<P>Note that the toris in the figure below are not identical. Arbitrary vertical slice of a torus are <em>not</em> Cassinian ovals, they are called <a href="../SpiricSections_dir/spiricSections.html">Spiric Sections</a>.</p>

<center>
<table cellpadding="10">
<tr>
<td><img src="cassinianOvalCut1.png" width="232" height="281"></td>
<td><img src="cassinianOvalCut2.png" width="232" height="280"></td>
</tr>
</table>
<p><img src="../Icons_dir/mmaIconSmall.gif" width="16" height="16"> <a href="torusCut.nb">graphics code</a>; <img src="../Icons_dir/gcf.gif" width="12" height="16"> <a href="cassin_oval_torus_cut.gcf">dynamic torus cut</a>
</p>
</center>

<p>Proof outline: start with a torus equation (Sqrt[x^2 + y^2] - d)^2 + z^2 == c^2. Eliminate the square root and regroup to one side. Replace d=a and c = b^2/(Sqrt[4]*a). Now do the same with cassian oval implicit equation Sqrt[(x-a)^2+y^2]*Sqrt[(x+a)^2+y^2]==b^2. Luckly, one sees that the two equations match without further algebra considering scale and rotation of the curve. (<img src="../Icons_dir/mmaIconSmall.gif" width="16" height="16"> <a href="torusCutProof.nb">detailed proof</a>)</p>


<p><img src="../Icons_dir/mmaIconSmall.gif" width="16" height="16"> <a href="cassinianOvalSurface.nb">Cassinian Oval as a surface.</a>.</p>


<hr>
<h2><a name="Related Web Sites">Related Web Sites</a></h2>

<p>see <b><A HREF="../Intro_dir/relatedHyperLinks.html">Generic Reference Page</A></b>.</p>

<p><a href="http://www-groups.dcs.st-andrews.ac.uk/%7Ehistory/Curves/Cassinian.html">MacTutor Famous Curve Index on Cassinian Ovals</a></p>


<p>
Visual Complex Analysis by Tristan Needham. p.60 - 63.
<A HREF="http://www.amazon.com/exec/obidos/ASIN/0198534469/xahhome-20">Amazon.com</A>
He also mention curves formed by the locus of points whose product of distances to n points are constant.
</p>


<pre>back to <a href="../specialPlaneCurves.html">Table of Contents</a></pre>
<hr>
<!-- footer xyz63 -->

<pre>Last updated: 2004-11.
&copy; copyright 1995-2004 by <a href="http://xahlee.org/PageTwo_dir/more.html">Xah Lee</a>. (<a href="mailto:xah@xahlee.org">xah@xahlee.org</A>)
http://xahlee.org/SpecialPlaneCurves_dir/specialPlaneCurves.html</pre>



</body>
</html>

